ECE DIGITAL LOGIC LECTURE 4: BINARY CODES Assistant Prof. Fareena Saqib Florida Institute of Technology Fall 2016, 01/26/2016

COMPLEMENTS OF NUMBERS Radix Complement Diminished Radix Complement Recap Using 10’s complement, subtract Given the two binary numbers X = and Y = , perform the subtraction (a) X - Y and (b) Y - X by using 2’s complements.

Agenda  Binary Codes  BCD- Binary coded decimal Code  Decimal Arithmetic  Grey Code  ASCII code  Error correction/Error detection  Binary storage and Registers

Binary Code  An n ‐ bit binary code is a group of n bits that assumes up to 2n distinct combinations of 1’s and 0’s, with each combination representing one element of the set that is being coded.  Although the minimum number of bits required to code 2n distinct quantities is n, there is no maximum number of bits that may be used for a binary code.  For example, the 10 decimal digits can be coded with 10 bits, and each decimal digit can be assigned a bit combination of nine 0’s and a 1.  Decimal Codes  Binary-coded Decimal Code (BCD)  Excess-3  2421  8,4,-2,-1  Gray Code  ASCII Character code  Error Correcting code

Binary-Coded Decimal Code  To represent the decimal digits by means of a code that contains 1’s and 0’s.  The code most commonly used for the decimal digits is the straight binary assignment listed.  Example  (396) 10  ( ) BCD Decimal SymbolBCD Digit

Binary-Coded Decimal Code: BCD Addition  Consider the addition of two decimal digits in BCD, together with a possible carry from a previous less significant pair of digits.  Since each digit does not exceed 9, the sum cannot be greater than = 19.  When the binary sum is equal to or less than 1001 (without a carry), the corresponding BCD digit is correct.  However, when the binary sum is greater than or equal to 1010, the result is an invalid BCD digit.  The addition of 6 = (0110)2 to the binary sum converts it to the correct digit and also produces a carry as required.

Binary-Coded Decimal Code: BCD Addition  Example of 1 digits number Addition  Addition of n-digit unsigned BCD numbers addition

Binary-Coded Decimal Code: BCD Addition  The representation of signed decimal numbers in BCD is similar to the representation of signed numbers in binary  We can use either the familiar signed ‐ magnitude system or the signed ‐ complement system.  It is customary to designate a plus with four 0’s and a minus with the BCD equivalent of 9, which is  The signed ‐ complement system can be either the 9’s or the 10’s complement, but the 10’s complement is the one most often used.  Consider the addition (+375) + (-240) = +135 In Signed-Complement System

Other Decimal Codes  The BCD code has weights of 8, 4, 2, and 1, which correspond to the power ‐ of ‐ two values of each bit.  The bit assignment 0110, for example, is interpreted by the weights to represent decimal 6 because 8 * * * * 0 = 6.  The bit combination 1101, when weighted by the respective digits 2421, gives the decimal equivalent of 2 * * * * 1 = 7.  Note that some digits can be coded in two possible ways in the 2421 code. For instance, decimal 4 can be assigned to bit combination 0100 or 1010, since both combinations add up to a total weight of 4.

Benefits of decimal codes  BCD adders add BCD values directly, digit by digit, without converting the numbers to binary.  The 2421 and the excess ‐ 3 codes are examples of self ‐ complementing codes. Such codes have the property that the 9’s complement of a decimal number is obtained directly by changing 1’s to 0’s and 0’s to 1’s.  For example, decimal 395 is represented in the excess ‐ 3 code as  The 9’s complement of 395 = 604 which is represented as , which is obtained simply by complementing each bit of the code (as with the 1’s complement of binary numbers).

Grey Code  The Gray code is used in applications in which the normal sequence of binary numbers generated by the hardware may produce an error or ambiguity during the transition from one number to the next.  Between any two consecutive numbers, the symbols have at most 1 bit different.  Gray code of 7 and 8 is 0100, 1100  Versus Binary of 7 and 8 is 0111, 1000.

Grey Code  Binary to Grey Code a) = 7 b) 1000 = (Try your self)  Grey Code to Binary Code a) = 7 b) 1100 = (Try your self)

ASCII Character Code  Many applications of digital computers require the handling not only of numbers, but also of other characters or symbols, such as the letters of the alphabet.  The standard binary code for the alphanumeric characters is the American Standard Code for Information Interchange (ASCII), which uses seven bits to code 128 characters.

Error Detecting Code  To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity.  A parity bit is an extra bit included with a message to make the total number of 1’s either even or odd.

Next Class  Digital Logic  Boolean Algebra